Rational Functions
A rational function is a function of
the form
h( x )
f ( x) 
g ( x)
where g (x)  0

Domain of a Rational Function
The domain of a rational function
h( x)
g ( x)
is the set of real numbers x so that g (x)

Example
If
5
y
x
the domain is all real numbers except x = 0
0
Domain of a Rational
Function
x2  x  6
f ( x) 
x2
If
then the domain is the set of all real
numbers except x = -2
If
2x
f ( x) 
x5
then the domain is the set of all real
numbers x > 5
Domain of a Rational
Function
If
f ( x) 
6 x
3x 2  2
then the domain is the set of all real
numbers, because the denominator of
this function is never equal to zero
Unbounbed Functions
A function f(x) is said to be unbounded in the
positive direction if as x gets closer to zero,
the values of f (x) gets larger and larger.
We write this as f (x)  
as x
0
(read f (x) approaches infinity as x goes to 0)
An Unbounbed Function
The function
f ( x) 
5
x2
below is unbounded
x
5
x2
-1/10
500
-1/100
5,0000
-1/1000
5,000,00
0
1/1000
5000,000
1/100
5,0000
1/10
500
Note that as x approaches 0, f (x)
becomes very large
Unbounbed Functions
Note that as x gets closer to zero, the values of f (x)
gets smaller and smaller.
We say f (x) is unbounded in the negative direction.
We write this as f (x)   as x 0
(read f (x) approaches negative infinity as x goes
to 0)
x

1
x2
-1/10
100
-1/100
10,000
-1/1000
1,000,000
1/1000
1,000,000
1/100
10,000
1/10
100
Unbounbed Functions
If as x gets closer to zero from the
left, the values of f (x) gets smaller
and smaller, and as x gets closer to
zero from the right, the values of f
(x) gets larger and larger,
we say f (x) is unbounded in both direction
Unbounbed Functions
and write this as f (x)  
as x
0
(read f (x) approaches positive or negative infinity
as x goes to 0)
Unbounbed Functions
f ( x) 
1
x
Vertical Asymptote
For any rational function f ( x)  h( x) in lowest terms and
g ( x)
a real number c so that h (c)  0 and g (c)= 0
the line x = c is called a vertical
The function
2x
f ( x) 
x4
has a vertical asymptote X = 4
asymptote
Vertical Asymptote
For the function
x2  x  6
f ( x) 
x2
the vertical
asymptote is x = -2
The function
f ( x) 
6 x
3x 2  2
has no vertical asymptote because the
denominator is never zero
`
Horizontal Asymptote
If the degree of the denominator of a rational
function f (x) = h (x) / g (x) is greater than or
equal to the degree of the numerator of the
rational function, then f (x) has a horizontal
asymptote.
Horizontal Asymptote
If the degree of the denominator is greater than
the of the degree of the numerator, then the
horizontal asymptote is y = 0
Example
f ( x) 
3x
x2  5
y = 0 is the horizontal asymptote
Horizontal Asymptote
If the degree of the denominator is equal to the
of the degree of the numerator, then the
horizontal asymptote is y = a where a is a
non zero real number
Example
f ( x) 
2x
x4
y = 2 is the horizontal asymptote
Horizontal Asymptote
Examples of horizontal asymptotes
2x
f ( x) 
x4
Horizontal asymptote is y = 2
f ( x) 
6 x
3x 2  2
Horizontal asymptote is y = 0
Slant Asymptote
If the degree of the numerator is greater to the
degree of the denominator, we have a slant
asymptote
Example of a function that has a slant asymptote
3x 3
f ( x)  2
x 5
Please review
problems solved in
class before doing
your homework